Problem: Simplify and expand the following expression: $ \dfrac{4k - 7}{k - 5}+\dfrac{k}{k + 9} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(k - 5)(k + 9)$ Multiply the first term by $\dfrac{k + 9}{k + 9}$ $ \begin{align*} \dfrac{4k - 7}{k - 5} \times \dfrac{k + 9}{k + 9} & = \dfrac{(4k - 7)(k + 9)}{(k - 5)(k + 9)} \\ & = \dfrac{4k^2 + 29k - 63}{(k - 5)(k + 9)}\end{align*} $ Multiply the second term by $\dfrac{k - 5}{k - 5}$ $ \begin{align*} \dfrac{k}{k + 9} \times \dfrac{k - 5}{k - 5} & = \dfrac{(k)(k - 5)}{(k + 9)(k - 5)} \\ & = \dfrac{k^2 - 5k}{(k + 9)(k - 5)}\end{align*} $ Now we have: $ = \dfrac{4k^2 + 29k - 63}{(k - 5)(k + 9)} + \dfrac{k^2 - 5k}{(k + 9)(k - 5)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4k^2 + 29k - 63 + k^2 - 5k}{(k - 5)(k + 9)} $ $ = \dfrac{5k^2 + 24k - 63}{(k - 5)(k + 9)}$ Expand the denominator: $ = \dfrac{5k^2 + 24k - 63}{k^2 + 4k - 45}$